Find the limit.
-$\lim _ { x \rightarrow 0 } \left( e ^ { 6 / x } + 5 x \right) ^ { x / 2 }$
A) 1
B) 0
C) $\mathrm { e } ^ { 3 }$
D) 3
E) $\infty$

Question

Quizplus · Accepted Answer

Rewrite the expression as 
\begin{align*}
 & \lim_{x	o 0} \big(e^{6/x}\big)^{x/2} \big(1+	frac{5x}{e^{6/x}}\big)^{x/2} \
=& \lim_{x	o 0} \big(e^{3}\big)^{3	frac{x}{6}} \big(1+	frac{5x}{e^{6/x}}\big)^{x/2}\
=& e^3\left(\lim_{x	o 0} \big(1+	frac{5x}{e^{6/x}}\big)^{e^{6/x}\cdot \frac{6}{5}\cdot(-5/x^2)\cdot (-x^2/2)}ight)\
=& e^3\left(\lim_{x	o 0} \big(1+	frac{5x}{e^{6/x}}\big)^{ -3 }ight)\
=& e^3. 
\end{align*}
Therefore, the answer is (C).

Find the Limit lim⁡x→0(e6/x+5x)x/2\lim _ { x \rightarrow 0 } \left( e ^ { 6 / x } + 5 x \right) ^ { x / 2 }limx→0​(e6/x+5x)x/2

Find the Limit $\lim _ { x \rightarrow 0 } \left( e ^ { 6 / x } + 5 x \right) ^ { x / 2 }$